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TransExample2011

TransExample2011 - 19 ECE 253a Digital Image Processing...

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# 19 ECE 253a Digital Image Processing Pamela Cosman 11/17/11 Quantitative Example of Energy Compaction and Decorrelation Let u be a 2 × 1 vector with mean zero: u = bracketleftBigg u (0) u (1) bracketrightBigg and covariance R u = parenleftBigg E [( u (0) - μ 0 )( u (0) - μ 0 )] E [( u (0) - μ 0 )( u (1) - μ 1 )] E [( u (0) - μ 0 )( u (1) - μ 1 )] E [( u (1) - μ 1 )( u (1) - μ 1 )] parenrightBigg = parenleftBigg E [ u (0) 2 ] E [ u (0) u (1)] E [ u (1) u (0)] E [ u (1) 2 ] parenrightBigg = parenleftBigg 1 ρ ρ 1 parenrightBigg for 0 < ρ < 1. From the expression for R u , the variances σ 2 u (0) = σ 2 u (1) = 1 that is, the total average energy of 2 is distributed equally between u (0) and u (1). The parameter ρ gives an indication of the correlation between u (0) and u (1). The correlation is by definition corr [ u (0) , u (1)] = E [ u (0) u (1)] σ u (0) σ u (1) = ρ So in this case, the off-diagonal elements of the covariance matrix are in fact exactly equal to the correlation, but that is only because the σ ’s are both equal to one. Now we transform u as follows: v = 1 2 parenleftBigg 3 1 - 1 3 parenrightBigg u
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